Chapter 6: Problem 30
Determine the solvability conditions for the nonhomogeneous boundralue problem: \(u^{\prime}(\pi / 4)=\beta\).
Chapter 6: Problem 30
Determine the solvability conditions for the nonhomogeneous boundralue problem: \(u^{\prime}(\pi / 4)=\beta\).
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Get started for freeThe Hermite polynomials, \(H_{n}(x)\), satisfy the following:
i. \(
Consider the problem: $$ \frac{\partial^{2} G}{\partial x^{2}}=\delta\left(x-x_{0}\right), \quad \frac{\partial G}{\partial x}\left(0, x_{0}\right)=0, \quad G\left(\pi, x_{0}\right)=0 $$ a. Solve by direct integration. b. Compare this result to the Green's function in part b of Problem \(31 .\) c. Verify that \(G\) is symmetric in its arguments.
Prove the double factorial identities: $$ \begin{gathered} (2 n) ! !=2^{n} n ! \\ (2 n-1) ! !=\frac{(2 n) !}{2^{n} n !} \end{gathered} $$
Prove that if \(u(x)\) and \(v(x)\) satisfy the general homogeneous boundary itions $$ \begin{aligned} &\alpha_{1} u(a)+\beta_{1} u^{\prime}(a)=0 \\ &\alpha_{2} u(b)+\beta_{2} u^{\prime}(b)=0 \end{aligned} $$ \(=a\) and \(x=b\), then $$ p(x)\left[u(x) v^{\prime}(x)-v(x) u^{\prime}(x)\right]_{x=a}^{x=b}=0 $$
The coefficients \(C_{k}^{p}\) in the binomial expansion for \((1+x)^{p}\) are given by $$ C_{k}^{p}=\frac{p(p-1) \cdots(p-k+1)}{k !} $$ a. Write \(C_{k}^{p}\) in terms of Gamma functions. b. For \(p=1 / 2\), use the properties of Gamma functions to write \(C_{k}^{1 / 2}\) in terms of factorials. c. Confirm your answer in part b. by deriving the Maclaurin series expansion of \((1+x)^{1 / 2}\)
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